I want to ask about infinitesimals and non-standard analysis. In physics we always use $\mathrm dx,~\mathrm dv,~\mathrm dt$ etc. as infinitesimal quantities. When we deduce equations in physics, when we set up integrals and in many other instances we use infinitesimals. For example in the first law of thermodynamics we have: $đQ=\mathrm dU+đW$.
Now when we ask teachers about what is $đQ$ here, they say it is an infinitesimal amount of thermal energy. So my question is, is it necessary to treat the whole non-standard analysis as an axiom for physics or do we have standard calculus explanation for physicist's use of $\mathrm dy/\mathrm dx$ as a quotient in all cases?